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## Linear Equations

We use the standard notation for a system of simultaneous linear equations:

 A x = b (2.4)

where A is the coefficient matrix, b is the right hand side, and x is the solution. In (2.4) A is assumed to be a square matrix of order n, but some of the individual routines allow A to be rectangular. If there are several right hand sides, we write

 A X = B (2.5)

where the columns of B are the individual right hand sides, and the columns of X are the corresponding solutions. The basic task is to compute X, given A and B.

If A is upper or lower triangular, (2.4) can be solved by a straightforward process of backward or forward substitution. Otherwise, the solution is obtained after first factorizing A as a product of triangular matrices (and possibly also a diagonal matrix or permutation matrix).

The form of the factorization depends on the properties of the matrix A. LAPACK provides routines for the following types of matrices, based on the stated factorizations:

• general matrices (LU factorization with partial pivoting):

A = PLU

where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n).

• general band matrices including tridiagonal matrices (LU factorization with partial pivoting): If A is m-by-n with kl subdiagonals and ku superdiagonals, the factorization is

A = LU

where L is a product of permutation and unit lower triangular matrices with kl subdiagonals, and U is upper triangular with kl+ku superdiagonals.
• symmetric and Hermitian positive definite matrices including band matrices (Cholesky factorization):

where U is an upper triangular matrix and L is lower triangular.
• symmetric and Hermitian positive definite tridiagonal matrices (L D LT factorization):

where U is a unit upper bidiagonal matrix, L is unit lower bidiagonal, and D is diagonal.

• symmetric and Hermitian indefinite matrices (symmetric indefinite factorization):

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with diagonal blocks of order 1 or 2.

The factorization for a general tridiagonal matrix is like that for a general band matrix with kl = 1 and ku = 1. The factorization for a symmetric positive definite band matrix with k superdiagonals (or subdiagonals) has the same form as for a symmetric positive definite matrix, but the factor U (or L) is a band matrix with k superdiagonals (subdiagonals). Band matrices use a compact band storage scheme described in section 5.3.3. LAPACK routines are also provided for symmetric matrices (whether positive definite or indefinite) using packed storage, as described in section 5.3.2.

While the primary use of a matrix factorization is to solve a system of equations, other related tasks are provided as well. Wherever possible, LAPACK provides routines to perform each of these tasks for each type of matrix and storage scheme (see Tables 2.7 and 2.8). The following list relates the tasks to the last 3 characters of the name of the corresponding computational routine:

xyyTRF:
factorize (obviously not needed for triangular matrices);

xyyTRS:
use the factorization (or the matrix A itself if it is triangular) to solve (2.5) by forward or backward substitution;

xyyCON:
estimate the reciprocal of the condition number ; Higham's modification [63] of Hager's method [59] is used to estimate |A-1|, except for symmetric positive definite tridiagonal matrices for which it is computed directly with comparable efficiency [61];

xyyRFS:
compute bounds on the error in the computed solution (returned by the xyyTRS routine), and refine the solution to reduce the backward error (see below);

xyyTRI:
use the factorization (or the matrix A itself if it is triangular) to compute A-1 (not provided for band matrices, because the inverse does not in general preserve bandedness);

xyyEQU:
compute scaling factors to equilibrate A (not provided for tridiagonal, symmetric indefinite, or triangular matrices). These routines do not actually scale the matrices: auxiliary routines xLAQyy may be used for that purpose -- see the code of the driver routines xyySVX for sample usage.

Note that some of the above routines depend on the output of others:

xyyTRF:
may work on an equilibrated matrix produced by xyyEQU and xLAQyy, if yy is one of {GE, GB, PO, PP, PB};

xyyTRS:
requires the factorization returned by xyyTRF;

xyyCON:
requires the norm of the original matrix A, and the factorization returned by xyyTRF;

xyyRFS:
requires the original matrices A and B, the factorization returned by xyyTRF, and the solution X returned by xyyTRS;

xyyTRI:
requires the factorization returned by xyyTRF.

The RFS (refine solution'') routines perform iterative refinement and compute backward and forward error bounds for the solution. Iterative refinement is done in the same precision as the input data. In particular, the residual is not computed with extra precision, as has been traditionally done. The benefit of this procedure is discussed in Section 4.4.

 Type of matrix Operation Single precision Double precision and storage scheme real complex real complex general factorize SGETRF CGETRF DGETRF ZGETRF solve using factorization SGETRS CGETRS DGETRS ZGETRS estimate condition number SGECON CGECON DGECON ZGECON error bounds for solution SGERFS CGERFS DGERFS ZGERFS invert using factorization SGETRI CGETRI DGETRI ZGETRI equilibrate SGEEQU CGEEQU DGEEQU ZGEEQU general factorize SGBTRF CGBTRF DGBTRF ZGBTRF band solve using factorization SGBTRS CGBTRS DGBTRS ZGBTRS estimate condition number SGBCON CGBCON DGBCON ZGBCON error bounds for solution SGBRFS CGBRFS DGBRFS ZGBRFS equilibrate SGBEQU CGBEQU DGBEQU ZGBEQU general factorize SGTTRF CGTTRF DGTTRF ZGTTRF tridiagonal solve using factorization SGTTRS CGTTRS DGTTRS ZGTTRS estimate condition number SGTCON CGTCON DGTCON ZGTCON error bounds for solution SGTRFS CGTRFS DGTRFS ZGTRFS symmetric/Hermitian factorize SPOTRF CPOTRF DPOTRF ZPOTRF positive definite solve using factorization SPOTRS CPOTRS DPOTRS ZPOTRS estimate condition number SPOCON CPOCON DPOCON ZPOCON error bounds for solution SPORFS CPORFS DPORFS ZPORFS invert using factorization SPOTRI CPOTRI DPOTRI ZPOTRI equilibrate SPOEQU CPOEQU DPOEQU ZPOEQU symmetric/Hermitian factorize SPPTRF CPPTRF DPPTRF ZPPTRF positive definite solve using factorization SPPTRS CPPTRS DPPTRS ZPPTRS (packed storage) estimate condition number SPPCON CPPCON DPPCON ZPPCON error bounds for solution SPPRFS CPPRFS DPPRFS ZPPRFS invert using factorization SPPTRI CPPTRI DPPTRI ZPPTRI equilibrate SPPEQU CPPEQU DPPEQU ZPPEQU symmetric/Hermitian factorize SPBTRF CPBTRF DPBTRF ZPBTRF positive definite solve using factorization SPBTRS CPBTRS DPBTRS ZPBTRS band estimate condition number SPBCON CPBCON DPBCON ZPBCON error bounds for solution SPBRFS CPBRFS DPBRFS ZPBRFS equilibrate SPBEQU CPBEQU DPBEQU ZPBEQU symmetric/Hermitian factorize SPTTRF CPTTRF DPTTRF ZPTTRF positive definite solve using factorization SPTTRS CPTTRS DPTTRS ZPTTRS tridiagonal estimate condition number SPTCON CPTCON DPTCON ZPTCON error bounds for solution SPTRFS CPTRFS DPTRFS ZPTRFS

 Type of matrix Operation Single precision Double precision and storage scheme real complex real complex symmetric/Hermitian factorize SSYTRF CHETRF DSYTRF ZHETRF indefinite solve using factorization SSYTRS CHETRS DSYTRS ZHETRS estimate condition number SSYCON CHECON DSYCON ZHECON error bounds for solution SSYRFS CHERFS DSYRFS ZHERFS invert using factorization SSYTRI CHETRI DSYTRI ZHETRI complex symmetric factorize CSYTRF ZSYTRF solve using factorization CSYTRS ZSYTRS estimate condition number CSYCON ZSYCON error bounds for solution CSYRFS ZSYRFS invert using factorization CSYTRI ZSYTRI symmetric/Hermitian factorize SSPTRF CHPTRF DSPTRF ZHPTRF indefinite solve using factorization SSPTRS CHPTRS DSPTRS ZHPTRS (packed storage) estimate condition number SSPCON CHPCON DSPCON ZHPCON error bounds for solution SSPRFS CHPRFS DSPRFS ZHPRFS invert using factorization SSPTRI CHPTRI DSPTRI ZHPTRI complex symmetric factorize CSPTRF ZSPTRF (packed storage) solve using factorization CSPTRS ZSPTRS estimate condition number CSPCON ZSPCON error bounds for solution CSPRFS ZSPRFS invert using factorization CSPTRI ZSPTRI triangular solve STRTRS CTRTRS DTRTRS ZTRTRS estimate condition number STRCON CTRCON DTRCON ZTRCON error bounds for solution STRRFS CTRRFS DTRRFS ZTRRFS invert STRTRI CTRTRI DTRTRI ZTRTRI triangular solve STPTRS CTPTRS DTPTRS ZTPTRS (packed storage) estimate condition number STPCON CTPCON DTPCON ZTPCON error bounds for solution STPRFS CTPRFS DTPRFS ZTPRFS invert STPTRI CTPTRI DTPTRI ZTPTRI triangular solve STBTRS CTBTRS DTBTRS ZTBTRS band estimate condition number STBCON CTBCON DTBCON ZTBCON error bounds for solution STBRFS CTBRFS DTBRFS ZTBRFS

Next: Orthogonal Factorizations and Linear Up: Computational Routines Previous: Computational Routines   Contents   Index
Susan Blackford
1999-10-01